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# Composition Of Functions & Inverse Of A Function

## Composite Functions

When two functions are combined in such a way that the output of one function becomes the input to another function, then this is referred to as composite function. Consider three sets X, Y and Z and let f : X → Y  and g: Y → Z.

According to this, under map f, an element  x ∈ X is mapped to an element y = f(x) ∈ Y which in turn is mapped by g to an element z ∈ Z  in such a manner that z = g(y) = g[(f(x)] .

This mapping comprising of mappings f and g is known as composition of mappings. It is denoted by gof . Therefore, we are mapping onto .

The composite function is denoted as:

$$\begin{array}{l}~~~~~~~~~\end{array}$$
( gof)(x) = g(f (X) )

Similarly, (fog) (x) = f (g(x))

So, to find (gof) (x), take f(x)  as argument for the function g.

Let us try to solve some questions based on composite functions.

 Let’s Work Out: Example 1: Given the function f(x) = 3x + 5  and g(x) = $$\begin{array}{l} 2x^3 \end{array}$$  .Find ( gof)(x) and ( fog)(x). Solution:  We know, (gof) (x) = g(f(x)) = g (3x+5) = $$\begin{array}{l} 2(3x+5)^3 \end{array}$$ Using Binomial Expansion, we have $$\begin{array}{l}(gof) (x) = 2 \left [ (3x)^{3} + 3.(3x)^{2}.5 + 3.(3x)(5)^{2}+(5)^3 \right ]\end{array}$$ $$\begin{array}{l}\Rightarrow (gof) (x) = 2 \left [ 27x^{3} + 135x^{2} + 225 x + 125 \right]\end{array}$$ $$\begin{array}{l}\Rightarrow (gof) (x) = 54x^{3} + 270x^{2} + 450 x + 250 \end{array}$$ Now, (fog)(x) = f $$\begin{array}{l}\left( g(x) \right)\end{array}$$ = f ( $$\begin{array}{l}2x^3\end{array}$$) = $$\begin{array}{l}3 (2x^{3}) + 5 \end{array}$$ $$\begin{array}{l}\Rightarrow (fog)(x) = 6x^{3} + 5\end{array}$$ Example 2: Let f(x) = $$\begin{array}{l}x^2\end{array}$$ and g(x) = $$\begin{array}{l} \sqrt{1 – x^2}\end{array}$$ .Find (gof)(x) and ( fog)(x) . Solution: (gof)(x) = g( f(x)) = g($$\begin{array}{l}x^2\end{array}$$) = $$\begin{array}{l} \sqrt{1 – (x^2)^2}\end{array}$$ = $$\begin{array}{l} \sqrt{1 – x^4} \end{array}$$ (fog) (x) = f(g(x)) = f( $$\begin{array}{l} \sqrt{1 – x^2} \end{array}$$) = ( $$\begin{array}{l} \sqrt{1 – x^2})^2\end{array}$$ = $$\begin{array}{l} 1 – x^2 \end{array}$$

## Inverse of a Function

Let f:X → Y. Now, let f represent a one to one function and y be any element of Y , there exists a unique element x ∈ X  such that y = f(x).Then the map

$$\begin{array}{l}~~~~~~~~~~\end{array}$$
$$\begin{array}{l} f^{-1}:f[X] \rightarrow X \end{array}$$

which associates to each element is called as the inverse map of f.

The function f(x) =

$$\begin{array}{l} x^5 \end{array}$$
and g(x) =
$$\begin{array}{l} x^{\frac 12} \end{array}$$
have the following property:

$$\begin{array}{l} f\left( g(x) \right) \end{array}$$
=
$$\begin{array}{l} f \left( x^{\frac 15} \right) \end{array}$$
=
$$\begin{array}{l} (x^{\frac 15} )^5 \end{array}$$
= x

$$\begin{array}{l} g\left( f(x) \right) \end{array}$$
=
$$\begin{array}{l} g \left( x^{5} \right) \end{array}$$
=
$$\begin{array}{l} (x^{5} )^{\frac 15} \end{array}$$
= x

Thus, if two functions f and g  satisfy

$$\begin{array}{l} f \left( g(x) \right) \end{array}$$
= x for every x in domain of f , then in such a situation we can say that the function f is the inverse of g  and g  is the inverse of f .

For finding the inverse of a function, we write down the function y as a function of x  i.e. y = f(x)   and then solve for x  as a function of y.

To have a better insight on the topic let us go through some examples.

 Let’s Work Out: Example: 1 If f(x) = $$\begin{array}{l}x^2\end{array}$$, g(x) = $$\begin{array}{l} \frac{x}{3} \end{array}$$   and h(x) = 3x+2 . Find out fohog(x). Solution:  $$\begin{array}{l} h\left( g(x) \right) \end{array}$$ = 3 $$\begin{array}{l} \left ( \frac x3 \right) \end{array}$$ + 2 = x + 2 fohog(x) = f $$\begin{array}{l} \left[ h\left(g(x)\right)\right] \end{array}$$ = $$\begin{array}{l} ( x+ 2)^2 \end{array}$$ This is the required solution. Example 2: Find the inverse of the function f(x) = $$\begin{array}{l} x^3 \end{array}$$ , x ∈ R. Solution: The given function f(x) = $$\begin{array}{l} x^3 \end{array}$$   is a one to one and onto function defined in the range → R  . Therefore, we can find the inverse of this function. To find the inverse, we need to write down this function as $$\begin{array}{l}~~~~~~~~~~~~~\end{array}$$ y = $$\begin{array}{l} x^3 \end{array}$$ In the above equation,y  is an arbitrary element from the range of f. If we solve for x from the above equation, we will get: $$\begin{array}{l}~~~~~~~~~~~~~\end{array}$$ x = $$\begin{array}{l} y^{\frac 13} \end{array}$$ This gives a function g:Y →X. This new function g can be defined as $$\begin{array}{l}~~~~~~~~~~~~~\end{array}$$ g(y) = $$\begin{array}{l} y^{\frac 13} \end{array}$$ This function g is the inverse of the function f since it’s domain is same as the range of the function f.Since g(y) = $$\begin{array}{l} y ^{\frac 12} \end{array}$$ , representing the independent variable with x , we get g(x) = $$\begin{array}{l} x^{\frac 13}\end{array}$$ = $$\begin{array}{l} f^{-1}(x) \end{array}$$..

### Practice Problems

1. Let S = {1, 2, 3}. Determine whether the functions f : S → S defined as f = {(1, 1), (2, 2), (3, 3)}. Find f–1, if it exists.
2. Find gof and fog, if f(x) = |x| and g(x) = |5x – 2|.
3. Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.

## Frequently Asked Questions – FAQs

### What are the conditions for the inverse function?

The conditions for two functions f and g to be inverses:
f(g(x)) =x for all x in the domain of g
g(f(x)) = x for all x in the domain of f
If f and g are inverses, composing f and g (in either order) creates the function that returns that input called the identity function for every input.

### What is the difference between inverse function and composite function?

A composite function is a function obtained when two functions are combined so that the output of one function becomes the input to another function.
A function f: X → Y is defined as invertible if a function g: Y → X exists such that gof = I_X and fog = I_Y. The function g is called the inverse of f and is denoted by f ^–1.

### What is the symbol of an inverse function?

The symbol of an inverse function is -1. That means f^-1 denotes the inverse of function f.

### Does every function have an inverse?

No, not every function has an inverse. If a function f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible.

### Why does a function have no inverse?

A function has no inverse if it does not follow the conditions to be invertible.